Thesis: Worked on sAb section

make

@@ -9,8 +9,8 @@ Presentation)	pdflatex "../presentation/presentation.tex" || exit 1
 		pdflatex "../presentation/presentation.tex" || exit 1
 		mv presentation.pdf ../
 		;;
-Symbols)	pdflatex "../presentation/symbols.tex" || exit 1
+Symbols)	pdflatex "../thesis/symbols.tex" || exit 1
-		pdflatex "../presentation/symbols.tex" || exit 1
+		pdflatex "../thesis/symbols.tex" || exit 1
 		scp symbols.pdf moerman@stitch.science.ru.nl:~/thesissymbols.pdf
 		ssh moerman@stitch.science.ru.nl 'pdf2svg thesissymbols.pdf thesissymbols.svg'
 		scp moerman@stitch.science.ru.nl:~/thesissymbols.svg ../

thesis/1_CategoryTheory.tex

@@ -21,12 +21,12 @@ Note that the collection of objects may be a proper class instead of a set, howe
 
 As the notation suggests maps can be thought of as functions, which is also the case in many examples.
 
-\begin{lemma}
+\begin{example}
 	The category $\Set$ has a objects sets, and as maps ordinary functions. Of course we then have the identity function $\id_X(x) = x$ and composition as usual.
-\end{lemma}
+\end{example}
-\begin{lemma}
+\begin{example}
 	The category $\Ab$ has a objects abelian groups, and the maps between two objects are exactly the grouphomomorphisms. We know that the identity function is indeed a grouphomomorphism, and composing two grouphomomorpisms, gives indeed a new grouphomomorphism.
-\end{lemma}
+\end{example}
 
 In fact almost any mathematical structure can be described as a category, we have: $\cat{Ring}$ for rings, $\cat{Vect}$ for $\R$-vectorspaces, $\cat{Set_{fin}}$ for finite sets, $\cat{Poset}$ for posets, etc. Of course we would also like to express relations between categories, for example every abelian group is also a set. This idea can be formulated by the notion of a functor.
 
@@ -40,9 +40,9 @@ In fact almost any mathematical structure can be described as a category, we hav
 	We normally do not write the index of $F_0$ or $F_1$, instead we wrtie $F$ for both functions.
 \end{definition}
 
-\begin{lemma}
+\begin{exlemma}
 	There is a category $\cat{Cat}$ with categories as objects, and functors as maps.
-\end{lemma}
+\end{exlemma}
 \begin{proof}
 	First we define the identity functor. Let $\cat{C}$ be a category, define $\id_\cat{C}(A) = A$ for any object $A \in \cat{C}$ and $\id_\cat{C}(f) = f$ for any map $f: A \to B$ in $\cat{C}$. Cleary we have $\id_\cat{C}(f) : \id_\cat{C}(A) \to \id_\cat{C}(B)$. Also $\id_\cat{C}(\id_A) = \id_A = \id_{\id_\cat{C}(A)}$ and $\id_\cat{C}(f \circ g) = f \circ g$. So indeed $\id_\cat{C}$ is a functor.
 

thesis/2_ChainComplexes.tex

@@ -44,9 +44,9 @@ In particular $\Delta^0$ is simply a point, $\Delta^1$ a line and $\Delta^2$ a t
 
 Note that if we have any continuous map $\sigma : \Delta^{n+1} \to X$ we can precompose with a face map to get $\sigma \circ \delta^i : \Delta^n \to X$. This will be used for defining the boundary operator. We can make pictures of this, and when concerning continuous maps $\sigma : \Delta^{n+1} \to X$ we will draw the images in the space $X$, instead of functions.
 
-\todo{CC: Make some pictures here}
+\todo{Ch: Make some pictures here}
 
-\todo{CC: Define free abelian group}
+\todo{Ch: Define free abelian group}
 
 We now have enough tools to define the singular chaincomplex of a space $X$.
 
@@ -64,4 +64,4 @@ This might seem a bit complicated, but we can pictures this in an intuitive way,
 	\caption{The boundary of a 2-simplex}
 \end{figure}
 
-\todo{CC: Proposition: $C(X) \in \Ch{\cat{Ab}}$}
+\todo{Ch: Proposition: $C(X) \in \Ch{\cat{Ab}}$}

thesis/3_SimplicialAbelianGroups.tex

@@ -0,0 +1,47 @@
+\section{Simplicial Abelian Groups}
+\label{sec:Simplicial Abelian Groups}
+
+\begin{definition}
+	We define a category $\DELTA$, where the objects are the finite ordinals $[n] = \{0, \dots, n\}$ and maps are monotone increasing functions.
+\end{definition}
+
+There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \emph{degeneracy} maps, defined as (resp.):
+
+$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
+$$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
+
+for each $n \in \N$. The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of these maps. So in a certain sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as follows.
+
+\begin{figure}[h!]
+	\label{fig:delta_cat}
+	\includegraphics{delta_cat}
+	\caption{The category $\DELTA$ with the face and degeneracy maps.}
+\end{figure}
+
+\todo{sAb: Epi-mono factorization}
+
+Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $F: \DELTA^{op} \to Ab$. And hence we can picture a simplicial abelian group as follows.
+
+\begin{figure}
+	\label{fig:simplicial_abelian_group}
+	\includegraphics{simplicial_abelian_group}
+	\caption{A simplicial abelian group.}
+\end{figure}
+
+Of course the maps $\delta_i$ and $\sigma_i$ satisfy certain equations, these are the so called \emph{simplicial equations}.
+\todo{sAb: Is \emph{simplicial equations} really a thing?}
+
+\begin{lemma}
+	The face and degeneracy maps in $\DELTA$ satisfy the simplicial equations, ie.:
+	\begin{align}
+		\delta_j\delta_i &= \delta_i\delta_{j-1}  \hspace{0.5cm} \text{ if } i < j,\\
+		\sigma_j\delta_i &= \delta_i\sigma_{j-1}  \hspace{0.5cm} \text{ if } i < j,\\
+		\sigma_j\delta_j &= \sigma_j\delta_{j+1} = \text{id},\\
+		\sigma_j\delta_i &= \delta_{i-1}\sigma_j  \hspace{0.5cm} \text{ if } i > j+1,\\
+		\sigma_j\sigma_i &= \sigma_i\sigma_{j+1}  \hspace{0.5cm} \text{ if } i \leq j.
+	\end{align}
+\end{lemma}
+\begin{proof}
+	By writing out the definitions given above.
+\end{proof}
+\todo{sAb: Say a bit more (because Mueger will not like this)}

thesis/DoldKan.tex

@@ -12,6 +12,9 @@
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{definition}[theorem]{Definition}
 \newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{example}[theorem]{Example}
+\newtheorem{exlemma}[theorem]{Example/Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
 
 \input{../thesis/preamble}
 
@@ -28,12 +31,17 @@ It is interesting because objects on the left hand side are considered to be alg
 $$ \pi_n(A) \iso H_n(N(A)) \text{ for all } n \in \N $$
 where $N: \sAb \to \Ch{\Ab}$ is one half of the equivalence.
 
+In the first section some definitions from category theory are given, because we will need them later on. Then in the second section we will discuss the first category involved in the correspondence, $\Ch{\Ab}$, the category of chain complexes. The third section then continues with the second category involved, $\sAb$, especially for this section we will need category theory. Then we will look at the coorespondence itself.
+
 \newpage
 \input{../thesis/1_CategoryTheory}
 
 \newpage
 \input{../thesis/2_ChainComplexes}
 
+\newpage
+\input{../thesis/3_SimplicialAbelianGroups}
+
 \newpage
 \listoftodos
 % \nocite{*}

thesis/preamble.tex

@@ -4,7 +4,6 @@
 \usepackage{color}
 \usepackage{listings}
 
-%\newcommand{\id}{\text{id}}
 \newcommand{\N}{\mathbb{N}}
 \newcommand{\Z}{\mathbb{Z}}
 \newcommand{\R}{\mathbb{R}}
@@ -12,6 +11,7 @@
 \newcommand{\Ab}{\cat{Ab}}
 \newcommand{\sAb}{\cat{sAb}}
 \newcommand{\Set}{\cat{Set}}
+\newcommand{\DELTA}{\cat{\Delta}}
 \newcommand{\Ch}[1]{\mathbf{Ch}(#1)}
 \newcommand{\Hom}[3]{\mathbf{Hom}_{#1}(#2, #3)}
 \newcommand{\id}{\mathbf{id}}

thesis/symbols.tex

@@ -0,0 +1,34 @@
+\documentclass[12pt]{amsproc}
+
+% a la fullpage
+\usepackage{geometry}
+\geometry{a4paper}
+\geometry{twoside=false}
+
+% Activate to begin paragraphs with an empty line rather than an indent
+\usepackage[parfill]{parskip}
+\setlength{\marginparwidth}{2cm}
+
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{lemma}[theorem]{Lemma}
+
+\input{../thesis/preamble}
+
+\title{Dold-Kan Correspondence}
+\author{Joshua Moerman}
+
+\begin{document}
+\maketitle
+
+\begin{definition}
+	We define a category $\DELTA$, where the objects are the finite ordinals $[n] = \{0, \dots, n\}$ and maps are monotone increasing functions.
+\end{definition}
+
+$$ [0] \to [1] \to [2] \to [3] \to \ldots $$
+$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
+$$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
+
+$$ A_0 \to A_1 \to A_2 \to A_3 $$
+
+\end{document}